Composite materials are widely used owing to their high strength and stiffness, low weight, and long fatigue life. Most composite materials are anisotropic and inhomogeneous, and thus exhibit significantly more complicated behaviors over their lifetimes than those of simple metals, particularly during repetitive cyclic loadings. For example, the fatigue behavior of a composite laminate is typically characterized by a combination of processes that have no counterparts in the behaviors of metals, such as matrix cracking, delamination, fiber-matrix debonding, and fiber breakage. The complex, multiple mechanisms by which composites can fatigue and fail has made it difficult to model, predict, and monitor fatigue damage in composite materials. Two prior approaches to evaluating fatigue life are the linear elastic strain energy life (W-N) model, and the hysteresis energy life model.
One approach is that of Shokrieh, M. M., and Taheri-Behrooz, F. 2006. A unified fatigue life model based on energy method, Composite Structures, 75:444-50. This fatigue life model was based on strain energy, and applied the model to on- and off-axis, unidirectional polymer-composite laminates subjected to tension-tension and compression-compression fatigue loading. Assuming that the stress-strain relation was elastic, the strain energy was normalized (with respect to the maximum monotonic strain energy, i.e., the product of the maximum monotonic stress and strain) to indirectly take into account the fiber orientation angle, yielding the relation:
                              Δ          ⁢                                          ⁢          W                =                                            1                              X                ⁢                                                                  ⁢                                  ɛ                                      1                    ⁢                    u                                                                        ⁢                          (                                                                                                                                            σ                                                      1                            ⁢                            max                                                                          ⁢                                                  ɛ                                                      1                            ⁢                            max                                                                                              -                                                                                                                                                          σ                                                  1                          ⁢                          min                                                                    ⁢                                              ɛ                                                  1                          ⁢                          min                                                                                                                                )                                +                                    1                              Y                ⁢                                                                  ⁢                                  ɛ                                      2                    ⁢                    u                                                                        ⁢                          (                                                                                                                                            σ                                                      2                            ⁢                            max                                                                          ⁢                                                  ɛ                                                      2                            ⁢                            max                                                                                              -                                                                                                                                                          σ                                                  2                          ⁢                          min                                                                    ⁢                                              ɛ                                                  2                          ⁢                          min                                                                                                                                )                                +                                    1                              S                ⁢                                                                  ⁢                                  ɛ                                      6                    ⁢                    u                                                                        ⁢                          (                                                                                                                                            σ                                                      6                            ⁢                            max                                                                          ⁢                                                  ɛ                                                      6                            ⁢                            max                                                                                              -                                                                                                                                                          σ                                                  6                          ⁢                          min                                                                    ⁢                                              ɛ                                                  6                          ⁢                          min                                                                                                                                )                                                          (        1        )            where ΔW is the normalized linear elastic strain energy; σ1max, σ2max, σ6max, σ1min, σ2min, σ6min are the maximum and minimum stress components; and ε1max, ε2max, ε6max, ε1min, ε2min, ε6min are the corresponding maximum and minimum strain components in the fiber directions. X, Y, and S are the maximum static strength in the lengthwise and crosswise directions, and the shear strength, respectively. ε1u, ε2u, ε6u are the ultimate strains in the monotonic test. In a tension-tension fatigue test, when the 0° fibers are under tension the 90° fibers are under compression. Therefore, the value of compressive strength should be used in Equation (1) to evaluate strain energy. Since the specimens used were thin, the value of the 90° edgewise compressive strength, which is equal to the 90° tensile strength, was used in Equation (1) to calculate strain energy. Taking advantage of the transformation relationships between the on- and off-axis stress and using the linear elastic assumption, Equation (1) becomes
                              Δ          ⁢                                          ⁢          W                =                                            1              +              R                                      1              -              R                                ⁢                                    Δσ              2                        ⁡                          (                                                                                          cos                      4                                        ⁢                    θ                                                        X                    2                                                  +                                                                            sin                      4                                        ⁢                    θ                                                        Y                    2                                                  +                                                                            sin                      2                                        ⁢                                          θcos                      2                                        ⁢                    θ                                                        S                    2                                                              )                                                          (        2        )            where R is the stress ratio
      (          R      =                        σ          min                          σ          max                      )    ,Δσ represents the stress range (Δσ=σmax−σmin), and θ denotes the angle between the lengthwise fibers and the load direction.
Measuring the experimental stress range then allows one to calculate the elastic strain energy density at different number of cycles at failure (Nf).
Another approach is that of Giancane, S., Panella, F. W., and Dattoma, V. 2010. Characterization of fatigue damage in long fiber epoxy composite laminates, International Journal of Fatigue, 32:46-53; and Petermann, J., and Plumtree, A. 2001. A unified fatigue failure criterion for unidirectional laminates, Composites Part A-Applied Science and Manufacturing, 32:107-18. This hysteresis energy life model is based on the hysteresis area under cyclic loading (H)—a quantity that represents dissipated energy. The hysteresis area increases during the fatigue process as the result of permanent deformation. The hysteresis energy has two components: one that is released during the unloading process, and a second that represents the damage energy responsible for the creation of new cracks and the propagation of the existing cracks. The cyclic hysteresis energy may be directly calculated from the experimental stress and strain measurements in real time as:
                    H        =                              ∑                          i              =              1                        n                    ⁢                                    (                                                σ                  i                                -                                  σ                                      i                    -                    1                                                              )                        ⁢                          (                                                ɛ                  i                                -                                  ɛ                                      i                    -                    1                                                              )                                                          (        3        )            where n is the number of points acquired per cycle during fatigue. The total hysteresis energy or accumulated fracture energy, wt, at failure is obtained by summing the hysteresis energy per cycle over the total number of cycles to failure, Nf.
                              w          t                =                              ∑                          j              =              1                                      N              f                                ⁢                      (                                          ∑                                  i                  =                  1                                n                            ⁢                                                (                                                            σ                      i                                        -                                          σ                                              i                        -                        1                                                                              )                                ⁢                                  (                                                            ɛ                      i                                        -                                          ɛ                                              i                        -                        1                                                                              )                                                      )                                              (        4        )            
Hysteresis energy variation in a composite material is considerably more complex than that in a metal. To the inventors' knowledge, there have been no prior reports relating the fatigue properties of a composite material to the hysteresis energy.
Natarajan, V., GangaRao, H. V. S., Shekar, V., 2005. Fatigue response of fabric-reinforced polymeric composites. Journal of Composite Materials 39, 1541-1559 disclose a study in which three glass fabric FRP composite material coupons and systems were tested at constant low amplitude fatigue loading. Experimental results suggested that for a given FRP material and load configuration, the energy loss per cycle due to fatigue damage was linear from about 10-90% of the fatigue life of the FRP composite material. The energy loss per cycle was reported to be a characteristic value of the constituent materials, and to vary with the induced fatigue strain levels by a power law. Based on the experimental results, the authors proposed a fatigue life prediction model to predict the useful life of FRP composites, with internal strain energy as the damage metric.
Shokrieh, M. M., Taheri-Behrooz, F., 2006. A unified fatigue life model based on energy method. Composite Structures 75, 444-450 disclose a fatigue life model based on the energy method, developed for unidirectional polymer composite laminates subjected to constant amplitude, tension-tension or compression-compression fatigue loading. The fatigue model was based on a static failure criterion, and was normalized to static strength in the fiber, matrix and shear directions. The model was said to predict fatigue life for unidirectional composite laminates over the range of positive stress ratios in various fiber orientation. The results of the model were reported to agree well with experimental data for carbon/epoxy and E-glass/epoxy unidirectional plies.
Varvani-Farahani, A., Haftchenari, H., Panbechi, M., 2007. An energy-based fatigue damage parameter for off-axis unidirectional FRP composites. Composite Structures 79, 381-389 discloses a model using an energy-based fatigue damage parameter to assess the fatigue damage of unidirectional glass-reinforced plastic (GRP) and carbon-fiber reinforced plastic (CFRP) composites. The proposed parameter was based on the physics and the mechanism of fatigue cracking within three damage regions: matrix (I), fiber-matrix interface (II), and fiber (III) as the number of cycles progressed. The parameter involved the shear and normal energies calculated from stress and strain components acting on these regions. In region I the damage initiated as microcracks within the matrix. For region II, the damage progressed along the matrix-fiber interface, leading to fiber fracture in region III. The fatigue damage model was said to successfully correlate the fatigue lives of unidirectional GRP and CFRP composites at various off-axis angles and stress ratios.
Giancane, S., Panella, F. W., and Dattoma, V. 2010. Characterization of fatigue damage in long fiber epoxy composite laminates. International Journal of Fatigue, 32, 46-53 disclosed a study of damage characterization of a GFRC laminate. Forty fatigue tests were executed and S-N curves traced. Two parameters were chosen to monitor damage evolution during each test: stiffness and dissipated energy per cycle. Three zones were observed in graphs of processed. The authors suggested that the most important structural transformations occurred only in the final part of life. A method for predicting the remaining life in a GFRC was proposed.
P. Reis et al., 2010. Fatigue Damage Characterization by NDT in Polypropylene/Glass Fibre Composites. Appl. Compos. Mater. 2011, 18, 409-419 discloses the results of a study on a glass-fiber-reinforced polypropylene composite in which the fatigue damage was investigated in terms of residual stiffness and temperature rise. Thermographic and acoustic emission techniques were used to aid the interpretation of the fatigue damage mechanisms. Different laminates were tested. For one series, all the layers had one of the two fiber directions oriented with the axis of the plate. For the other two series the layer distribution was obtained with differing laminate orientations with respect to the axis of the sheet. The authors concluded that the residual stiffness and temperature rise could be used to predict final failure of a structure or component. Thermographic techniques were said to allow prediction of the site where failure would occur.
L. Toubal et al., 2006. Damage evolution and infrared thermography in woven composite laminates under fatigue loading. International Journal of Fatigue 28, 1867-1872 discloses an analytical model based on cumulative damage for predicting the damage evolution in composite materials. The model was compared with experimental data from a carbon/epoxy composite fatigued under tension-tension load. Fatigue tests of were monitored with an infrared thermography system. By analyzing the temperature of the external surface during the application of cyclic loading, it was said to be possible to evaluate damage evolution. The model was said to agree well with experimental data, and to be useful for predicting the evolution of damage in composites.
It is substantially easier to model and monitor fatigue and failure in a single-phase material, such as a metal, than in a multi-phase material, such as a composite. Results obtained in the former cannot in general be extrapolated to the latter.
We have previously reported that for two metals, Al 6061-T6 and SS 304, by tallying the entropy production up to the fracture point, one arrives at a unique material property—a so-called fracture fatigue entropy (FFE)—that is independent of geometry, load, and frequency. A necessary and sufficient condition for the metal to fracture due to cyclic loading is that its accumulated production of entropy should reach a certain level. See Naderi M, Amiri M, Khonsari M M. On the thermodynamic entropy of fatigue fracture. Proc. Roy. Soc. A. 2010; 466:423-438. See also co-pending patent application Ser. No. 12/898,100, the complete disclosure of which is incorporated by reference.
However, methods that may be successful in predicting the failure of metals cannot, in general, be successfully extrapolated to predicting the failure of composites. While metals are typically homogeneous and isotropic, most composites are neither homogeneous nor isotropic. The behaviors of damage stages and entropy accumulation are more complex in nonhomogeneous materials such as composites. Instead of a continuum damage model, which provides a good description of the development and growth of microcracks and macrocracks in many metals, in a composite material damage increases in a more complex manner, and sometimes in a sudden manner, due to matrix cracking, fiber/matrix delamination, breaking of fibers at weak interfaces, and other processes that do not have direct analogs in a single, homogeneous metal phase. In a composite there is typically a first stage that is characterized by low losses from strain energy, followed by a second stage in which damage gradually accumulates, and then a third and final phase in which there is a rapid increase in damage up to failure as the stronger fibers begin to break. Due to complexities such as these, it has been more difficult to model damage and failure in composites than in metals. Models that may work well for metals would not, in general, be expected to be successful in nonhomogeneous materials such as composites.
Many previous models of fatigue failure for composites have been based on the assumption that cycles of stress have constant amplitude, or other simplified loading conditions. There remains an unfilled need for a general method for monitoring and predicting fatigue failure for composite materials under more variable, “real world” loading conditions.